Spin Networks and Recoupling in Loop Quantum Gravity

TL;DR

This paper explores the use of spin-network basis and recoupling theory in loop quantum gravity to derive explicit expressions for quantum operators and establish orthonormality of states.

Contribution

It introduces a recoupling theory approach to explicitly compute eigenvalues and define a scalar product in loop quantum gravity.

Findings

Derived explicit eigenvalues of the quantum volume operator

Established a scalar product making area and volume operators symmetric

Proved orthonormality of spin network states

Abstract

I discuss the role played by the spin-network basis and recoupling theory (in its graphical tangle-theoretic formulation) and their use for performing explicit calculations in loop quantum gravity. In particular, I show that recoupling theory allows the derivation of explicit expressions for the eingenvalues of the quantum volume operator. An important side result of these computations is the determination of a scalar product with respect to which area and volume operators are symmetric, and the spin network states are orthonormal.